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A Stereo Image. The space of all stereo images. The space/geometry of all stereo/epipolar images/cameras. S. Seitz, J. Kim, The Space of All Stereo Images , IJCV 2002 / ICCV 2001. T. Pajdla, Epipolar Geometry of Some Non-Classical Cameras , Slovenian
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The spaceof all stereo images The space/geometry of all stereo/epipolar images/cameras • S. Seitz, J. Kim,The Space of All Stereo Images, IJCV 2002 / ICCV 2001. • T. Pajdla, Epipolar Geometry of Some Non-Classical Cameras, Slovenian • Computer Vision Winter Workshop, 2001. or The space (geometry) of ray sets (cameras) that allow row-based stereo analysis
Outline 2) Generalized / non-classical camera 1) Regular stereo 3) Stereo with generalized cameras (an example) 4) When can we compute stereo from generalized cameras?
Stereoscopic illusion Real world Stereo Illusions
Stereoscopic Imaging • Key property: horizontal parallax • Enables stereoscopic viewing and stereo algorithms Thank you, Steve Seitz
Capturing stereo pairs Left camera Right camera
2) Color glasses p 0 3) Polarized glasses • Temporally synchronized • screen and glasses Display different image for each eye 1) Separating with vertical paper
Parallax and disparity Aligned images Original images L 1 row R Z=1 Z=2 Z=0 Z=0 Z=1 Z=2 L L Z R R Disparity (1D) Alignment Measure depth with respect to this plane Parallax (3D)
Stereo Algorithms Photogrammetry (generating maps)
Stereoscopic Imaging • Key property: horizontal parallax • Enables stereoscopic viewing and stereo algorithms Thank you, Steve Seitz
RectificationorWhy we can focus on 1-row Line corresponds to line (“epipolar lines”) Rectified images: epipolar lines are image rows Homography ? ?
Outline 2) Generalized / non-classical camera 1) Regular stereo 3) Stereo with generalized cameras (an example) 4) When can we compute stereo from generalized cameras?
Geometric camera model We model camera as Mapping: world points image points (pixels) • We model only the “projection” operation • We do not model: light, color, lens blurring, etc.
Important property: every ray from the ray set of the camera projects to one point Example: pinhole camera model(usual camera) This projection operation is commonly described by a 3x4 projective matrix. For our purposes the following is more convenient: • A pinhole camera is defined by: • set of rays (starting from the camera center) • mapping from this rays to the image plane
A generalized camera maps rays to image points For our purposes camera set of rays Generalization of the camera model Classical camera • One center of projection • Image surface (film) is planar Generalized (ray-projective) camera • Multiple centers of projection (origins of rays) • Image surface is arbitrary
Xerox machine(non-classical cameras, example 1) As a multi-prospective camera:
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Pushbroom camera(non-classical cameras, example 2) • 1D projective sensor • … translating Advantage: large field of view in one dimension
y x Pushbroom camera(non-classical cameras, example 2) The generalized camera model: X direction - parallel projectionY direction - perspective projection NotionGenerator – the set of all ray origins For other cameras, generator can be a 2D surface
thanks to Shmuel Peleg, Hebrew University of Jerusalem Pushbroom camera(non-classical cameras, example 2) Imaging process Images from usual camera t t+1 t+2 Image from generalized camera Virtual generalized camera = device (usual camera) + software
t Non-classical cameras • Implementation through cuts of 3D video arrays • Take images while moving a usual camera • Stack them into 3D array • Take a cut along the “time” dimension
thanks to Shmuel Peleg, Hebrew University of Jerusalem Circular projective camera(non-classical cameras, example 3) Move “1D sensor” along a circlerecord on a cylinder Advantage: complete 360° horizontal view
Circular projective camera(non-classical cameras, example 3) The generalized camera model: Note: Generator is a circle
Outline 2) Generalized / non-classical camera 1) Regular stereo 3) Stereo with generalized cameras (an example) 4) When can we compute stereo from generalized cameras?
thanks to Steve Seitz, University of Washington Generalized stereo: an example Inward-facing camera, moving around an object
Output: 2 symmetric cuts of 3D video array thanks to Steve Seitz, University of Washington Images for both eyes • Input: video sequence
thanks to Steve Seitz, University of Washington Results: red-blue stereo image
thanks to Steve Seitz, University of Washington Results: 3D reconstruction Using usual algorithm (built for usual camera)with non-classical images
Ray geometry How does the set of rays look? Pixel = ray There is a “blind” area in the center of the scene
Ray geometry What rays go through a point in the scene?How disparity depends on depth?
Outline 2) Generalized / non-classical camera 1) Regular stereo 3) Stereo with generalized cameras (an example) 4) When can we compute stereo from generalized cameras?
Choose “ground” plane Z=0 L R PARRALAX / DISPARITY L R Z=0 Z=1 Z=2 Parallax and disparity in Cyclographs(review)
D The generalized camera model • D - image surface; P – ray space • A view (camera) is a function V: DP • Does not include: • Multiple rays to 1-image point • Curved light paths (mirror, lens)
A row (row-continuity) A row is the set of points in one view image that corresponds to a ray of the other view. For rectified images: rows are image rows Infinitesimality: • Row has width 0 • Row has no holes
D1 D2 Rays V(u1,v1) and V(u2,v2) intersect v1=v2 Stereo: basic constraints (u2,v2) (u1,v1) row v =v2 row v =v1
Basic stereo constraints + row-continuity Surfaces V(*,v1) and V(*,v2) , where v1, and v2 are corresponding rows, intersect in a surface(not a curve) D1 D2 row v =v1 (u2,v2) (u1,v1) row v =v2
Example: The intersection of epipolar planes The red plane ”intersects” the blue plane
Ruled surfaces: examples • Generalized cylinder • Generalized cone
The most important slide Doubly ruled surface • Left camera “ruling” the scene • Right camera “ruling” the scene
A hyperboloid Doubly Ruled surfaces: examples A plane
hyperboloid hyperbolic paraboloid plane Theorem (D. Hilbert ):The only doubly ruled surfaces are:
SUMMARY The space (geometry) of ray sets (cameras) that allow row-based stereo analysis are doubly ruled surfaces
Stereo Image Spiral mirror acquiring right eye panorama Spiral mirror acquiring left eye panorama Optical center viewing circle